To allow anonymous users to add items to a SharePoint list without giving them edit permissions to your site you can run the following PowerShell command.
add-pssnapin microsoft.sharepoint.powershell
$siteurl = "[your site URL]"
$listname = "[your list name]"
$a = (get-spweb $siteurl).lists | where title -eq $listname
$a.AnonymousPermMask64 = "ViewListItems, AddListItems, OpenItems, ViewVersions, ViewFormPages, Open, UseClientIntegration, UseRemoteAPIs"
Just enter the URL of you site and the name of the list in the appropriate values.